INSTABILITY OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS OF FIFTH ORDER

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Vol. 22, No. 1 (2016), pp. 83–91.

INSTABILITY OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS OF FIFTH ORDER

Cemil Tunc¸¹ and Muzaffer Ates¸²

1Department of Mathematics Faculty of Sciences Yüzüncü Yıl University 65080, Van Turkey

2Department of Electrical & Electronics Engineering Yüzüncü Yıl University 65080, Van Turkey

Abstract. This paper establishes certain sufficient conditions to guarantee the non- existence of periodic solutions for a class of nonlinear vector differential equations of fifth order. With this work, we extend and improve two related results in the literature from scalar cases to vectorial cases. An example is given to illustrate the theoretical analysis made in this paper.

Key words and Phrases: Instability, Lyapunov function, nonlinear, vector differential equation, fifth order.

Abstrak. Makalah ini membahas kondisi cukup untuk menjamin non-eksistensi dari solusi periodek untuk suatu kelas vektor nonlinear dari persamaan differensial orde lima. Dalam pembahasan makalah ini, dua hasil terkait yang ada dalam liter- atur telah dikembankan dan diperluas dari kasus skalar menjadi kasus vektor. Se- buah contoh diberikan untuk mengilustrasikan analisis teoritis yang dibahas dalam makalah ini.

Kata kunci: .

1. Introduction

It is well known that in applied sciences, some practical problems concerning mechanics, the engineering technique fields, economy, control theory, physics, chemistry, biology, medicine, atomic energy, information theory, etc. are associated with certain differential equations of higher order. Here, we would not like to give the details of them. By this time, perhaps, the most effective basic tool in the

2000 Mathematics Subject Classification: 34D05, 34D20..

Received: 18-02-2015, revised: 25-04-2016, accepted: 26-04-2016.

83

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literature to investigate the qualitative behaviors of certain differential equations whose orders are more than two is the Lyapunov’s direct method. Hence, Lyapunov functions have been successfully used and are still being used to discuss stability, instability, existence and non-existence of periodic solutions, etc. of differential equations whose orders are more than two.

In (2012), Tejumola [10] investigated the non-existence of periodic solutions to the following nonlinear scalar differential equations of fifth order

x⁽⁵⁾+ φ1(...

x )x⁽⁴⁾+ φ2(¨x)...

x + φ3( ˙x)¨x + φ4( ˙x) + φ5(x) = 0 (1) and

x⁽⁵⁾+ b1x⁽⁴⁾+ ψ2(¨x)...

x + ψ3( ˙x)¨x + ψ4( ˙x) + ψ5(x) = 0. (2) The author established certain sufficient conditions which guarantee that Eq.(1) and Eq.(2) have no non-trivial periodic solutions of whatever period with the aid of the Lyapunov’s direct method.

In this direction, in recent years, Ezeilo [1]-[3], Li and Duan [6], Li and Yu [7], Sadek [8], Sun and Hou [9], Tejumola [10], Tunc [11]-[13], [15], Tunc and Erdo- gan [16], Tunc and Karta [17], Tunc and S¸evli [18], etc., continued to discuss the existence, non-existence of periodic solutions and instability of solutions to certain nonlinear scalar and vector differential equations of fifth order by the Lyapunov’s second method. These researchers obtained many new and considerable results concerning to the mentioned topics. It should be noted that throughout these mentioned papers, the Lyapunov’s direct method has been used as a basic tool to investigate the main results thereof.

In this paper, we focus on the work of Tejumola [10]. Namely, instead of Eq.(1) and Eq.(2), we consider their following vectorial forms:

X⁽⁵⁾+ Φ1(...

X)X⁽⁴⁾+ Φ2( ¨X)...

X + Φ3( ˙X) ¨X + Φ4( ˙X) + Φ5(X) = 0 (3) and

X⁽⁵⁾+ AX⁽⁴⁾+ Ψ₂( ¨X)...

X + Ψ3( ˙X) ¨X + Ψ₄( ˙X) + Ψ₅(X) = 0, (4) respectively, where X ∈ <ⁿ; A is a constant n × n-symmetric matrix; Φ₁, Φ₂, Φ₃, Ψ₂ and Ψ₃ are n × n-symmetric continuous matrix functions; Φ₄: <ⁿ → <ⁿ, Φ₅ : <ⁿ → <ⁿ, Ψ₄ : <ⁿ → <ⁿ, Ψ₅ : <ⁿ → <ⁿ with Φ₄(0) = Φ₅(0) = Ψ₄(0) = Ψ₅(0) = 0 are continuous functions and so constructed such that the uniqueness theorem is valid.

Instead of Eq.(3) and Eq.(4), we consider their equivalent differential systems:

X = Y, ˙˙ Y = Z, ˙Z = W, W = U,˙

U = −Φ˙ ₁(W )U − Φ₂(Z)W − Φ₃(Y )Z − Φ₄(Y ) − Φ₅(X), (5) and

X = Y, ˙˙ Y = Z, ˙Z = W, W = U,˙

U = −AU − Ψ˙ ₂(Z)W − Ψ₃(Y )Z − Ψ₄(Y ) − Ψ₅(X), (6)

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respectively.

For the sake of the brevity, we assume that the symbol J_Φ₁(W ), J_Φ₂(Z), J_Φ₄(Y ), J_Φ₅(X), J_Ψ₂(Z), J_Ψ₃(Y ), J_Ψ₄(Y ) and J_Ψ₅(X) denote the Jacobian matrices corresponding to Φ1, Φ2, Φ4, Φ5, Ψ2, Ψ3, Ψ4 and Ψ5, respectively. In addition, it is assumed, as basic throughout the paper, that these Jacobian matrices exist and are continuous and symmetric.

We establish two new theorems on the non-existence of periodic solutions of Eq.(3) and Eq.(4). This paper is inspired by the results established in the aforementioned papers, Tunc [14] and in the literature. Our aim is to generalize and improve the results of Tejumola [10, Theorem 3, 5]. This paper has also a contribution to the subject in the literature, and it may be useful for researchers who work on the qualitative behaviors of solutions. The equation considered and the assumptions to be established here are different from those in aforementioned papers and in the literature.

The symbol hX, Y i corresponding to any pair X, Y in <ⁿ stands for the usual scalar product Pn

i=1xiyi and λi(A), (A = (aij)), (i, j = 1, 2, . . . , n) are the eigenvalues of the n × n-symmetric matrix A and the matrix A = (aij) is said to be positive definite if and only if the quadratic form X^TAX is positive definite, where X ∈ <ⁿ and X^T denotes the transpose of X.

Consider the linear constant coefficient differential equation of fifth order:

x⁽⁵⁾+ a1x⁽⁴⁾+ a2

...x + a3x + a¨ 4x + a˙ 5x = 0, (7) where a1, a2, . . . , a5 are some real constants. It can be seen from Tejumola [10]

that if either of the conditions

(i) a16= 0, sgn a1= sgn a5, a3 sgn a1< 0 and

(ii) a2< 0, a4> 0

holds, then Eq.(7) has no non-trivial periodic solutions of any period. It should also be noted that these odd and even subscripts features run through the generalized criteria obtained for the non-linear equations studied here.

2. Main Results

The following lemma plays a key role in proving our main results.

Lemma 2.1. (Horn and Johnson [4]) Let A be a real n × n-symmetric matrix and a⁰ ≥ λi(A) ≥ a, (i = 1, 2, . . . , n)

where a⁰ and a are some positive constants. Then a⁰hX, Xi ≥ hAX, Xi ≥ a hX, Xi and

a⁰²hX, Xi ≥ hAX, AXi ≥ a²hX, Xi .

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Our first main result is the following theorem.

Theorem 2.2. In addition to the basic assumptions imposed on Φ1, Φ2, Φ3, Φ4

and Φ5that appearing in Eq.(3), we assume that there are positive constants a1, a4

and a₅ and a₃(< 0) such that the following assumptions hold:

λ_i(Φ₁(W )) ≥ a₁, λ_i(Φ₃(Y )) ≤ a₃, λ_i(J_Φ₄(Y )) ≥ a₄, Φ₅(0) = 0, Φ₅(X) 6= 0, when X 6= 0, λ_i(J_Φ₅(X)) ≥ a₅. Then, Eq.(3) has no non-trivial periodic solution of whatever period.

Remark 1. There is no restriction on matix function Φ2 except Φ2 is an n × n-symmetric continuous matrix function.

Remark 2. To complete the proof of Theorem 1, subject to the assumptions of Theorem 1, we have to show that there exists a Lyapunov function V = V (X, Y, Z, W, U ), which satisfies the following Krasovskiis [5] criteria:

(K₁): In every neighborhood of (0, 0, 0, 0, 0) there exists a point (ξ, η, ζ, µ, ρ) such that V (ξ, η, ζ, µ, ρ) > 0;

(K2): the time derivative ˙V along solution paths of the system (5) is positive semi-definite;

(K3): the only solution (X, Y, Z, W, U ) = (X(t), Y (t), Z(t), W (t), U (t)) of system (5) which satisfies ˙V = 0, (t ≥ 0), is the trivial solution (0, 0, 0, 0, 0).

These properties guarantee that Eq.(3) has no non-trivial periodic solution of any period.

It should be noted that a similar discussion can be made for our second main result, Theorem 2.

Proof. To prove Theorem 1, we define a Lyapunov function V = V (X, Y, Z, W, U ):

V =

Z 1 0

hΦ₁(σW )W, Zi dσ + Z 1

0

hσΦ₂(σZ)Z, Zi dσ −1

2hW, W i +

Z 1 0

hΦ₄(σY ), Y i dσ + hΦ₅(X), Y i + hU, Zi . (8) First, it is easy to see from (8) that

V (0, 0, 0, 0, 0) = 0.

In view of the estimates _∂σ^∂ Φ₄(σY ) = J_Φ₄(σY )Y and Φ₄(0) = 0, it follows, on integrating both sides from σ₁= 0 to σ₁= 1, that

Φ4(Y ) = Z 1

0

JΦ₄(σ1Y )Y dσ1. Hence, we have

Z 1 0

hΦ4(σY ), Y i dσ = Z 1

0

Z 1 0

hσ1JΦ4(σ1σ2Y )Y, Y i dσ2dσ1

≥ Z 1

0

Z 1 0

hσ1a4Y, Y i dσ2dσ1≥ 1

2a4hY, Y i

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by the assumption λ_i(J_Φ₄(Y )) ≥ a₄.

Using this estimate, one can easily see that V (0, ε, 0, 0, 0) ≥ 1

2a4hε, εi = 1

2a4kεk²> 0

for all arbitrary ε 6= 0, ε ∈ <ⁿ, which verifies the property (K1) of Krasovskii [5].

Let (X, Y, Z, W, U ) = (X(t), Y (t), Z(t), W (t), U (t)) be an arbitrary solution of system (5). Differentiating the Lyapunov function V with respect to the time t along this solution, we get

d

dtV = d dt

Z 1 0

hΦ1(σW )W, Zi dσ + hJΦ₅(X)Y, Y i + d dt

Z 1 0

hσΦ2(σZ)Z, Zi dσ

− hΦ3(Y )Z, Zi + d dt

Z 1 0

hΦ4(σY ), Y i dσ − hΦ1(W )U, Zi

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− hΦ2(Z)W, Zi − hΦ4(Y ), Zi . It can be checked that

d dt

Z 1 0

hΦ1(σW )W, Zi dσ = Z 1

0

hΦ1(σW )W, W i dσ + Z 1

0

σ ∂

∂σhΦ1(σW )U, Zi dσ +

Z 1 0

hΦ1(σW )U, Zi dσ

= σ hΦ1(σW )U, Zi ¹₀ +

Z 1 0

hΦ1(σW )W, W i dσ

= hΦ1(W )U, Zi + Z 1

0

hΦ1(σW )W, W i dσ,

d dt

Z 1 0

hσΦ2(σZ)Z, Zi dσ = Z 1

0

σ ∂

∂σhΦ2(σZ)W, Zi dσ + Z 1

0

hσΦ2(σZ)W, Zi dσ

= σ²hΦ2(σZ)W, Zi |¹₀= hΦ2(σZ)W, Zi and

d dt

Z 1 0

hΦ4(σY ), Y i dσ = Z 1

0

σ hJΦ4(σY )Z, Y i dσ + Z 1

0

hΦ2(σY ), Zi dσ

= Z 1

0

σ ∂

∂σhΦ4(σY ), Zi dσ + Z 1

0

hΦ4(σY ), Zi dσ

= σ hΦ4(σY ), Zi

¹₀ = hΦ4(Y ), Zi .

Combining the last three estimates into (9) and in viewing of Lemma and the assumptions of Theorem 1, λ_i(Φ₁(W )) ≥ a₁> 0, λ_i(Φ₃(Y )) ≤ a₃< 0, λ_i(J_Φ₅(X)) ≥

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a₅> 0, we obtain

V = hJ˙ Φ₅(X)Y, Y i − hΦ3(Y )Z, Zi + Z 1

0

hΦ1(σW )W, W i dσ ≥ 0, which verifies the property (K₂) of Krasovskii [5].

Thus, the assumptions of Theorem 1 imply that ˙V (t) ≥ 0 for all t ≥ 0, that is, V is positive semi-definite. Finally, ˙˙ V = 0, (t ≥ 0), necessarily implies that Y = 0 for all t ≥ 0, and Z = ˙Y = 0, W = ¨Y = 0, W =˙ ...

Y = 0 for all t ≥ 0 so that X = ξ, (ξ 6= 0 is a constant vector) , Y = Z = W = U = 0.

From the last estimate and system (5), we have Φ₅(ξ) = 0 which necessarily implies that ξ = 0 since Φ₅(0) = 0. Then

X = Y = Z = W = U = 0 for all t ≥ 0,

which verifies the property (K₃) of Krasovskii [5]. Therefore, the Lyapunov function has the entire criteria of Krasovskii [5] if the assumptions of Theorem 1 hold. Thus, the basic properties of the Lyapunov function which were shown above, prove that system (5) has no non-trivial periodic solutions of whatever period. Since system (5) is equivalent to Eq.(3), this completes the proof of Theorem 1.

Example. As a special case of system (5), we choose Φ1, Φ2, Φ3, Φ4 and Φ5

as the following:

Φ1(W ) =

2 + (1 + w²)⁻¹ 1 1 2 + (1 + w²)⁻¹

,

Φ2(Z) =

−2 − z² 1 1 −2 − z²

,

Φ3(Z) =

−4 − y² 1 1 −4 − y²

,

Φ₄(Z) =

2y + arctan y 2y + arctan y

, Φ₄(0) = 0,

Φ₅(Z) =

3x + arctan x 3x + arctan x

, Φ₅(0) = 0.

It follows from Φ₄and Φ₅ that JΦ₄(Y ) =

2 + (1 + y²)⁻¹ 0 0 2 + (1 + y²)⁻¹

and

λi(JΦ5(X)) =

3 + (1 + x²)⁻¹ 0 0 3 + (1 + x²)⁻¹

. Then, respectively, we get

λ1(Φ1(W )) = 1 + 1

1 + w², λ2(Φ1(W )) = 3 + 1 1 + w²,

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λi(Φ1(W )) ≥ 1 = a1,

λ1(Φ2(Z)) = −1 − z², λ2(Φ2(Z)) = −3 − z²,

λi(Φ2(Z)) < 0,

λ1(Φ3(Y )) = −3 − y², λ2(Φ3(Y )) = −5 − y²,

λi(Φ3(Y )) ≤ −3 = a3,

λ₁(J_Φ₄(Y )) = 2 + 1

1 + y², λ₂(J_Φ₄(Y )) = 2 + 1 1 + y², λ_i(J_Φ₄(Y )) ≥ 2 = a₄,

λ1(JΦ₅(X)) = 3 + 1

1 + x², λ2(JΦ₅(X)) = 3 + 1 1 + x², λi(JΦ₅(X)) ≥ 3 = a5.

Thus, it is shown that all the assumptions of Theorem1 hold.

Our second main result is the following theorem.

Theorem 2.3. In addition to the basic assumptions imposed on A, Ψ₂, Ψ₃, Ψ₄ and Ψ₅ that appearing in Eq.(4), we assume that there are constants b₁(> 0), b₂(<

0), b₃(< 0), b₄(> 0) and b₅(> 0) such that the following conditions hold:

λ_i(A) ≥ b₁, λ_i(Ψ₂(Z)) ≤ b₂, λ_i(Ψ₃(Y )) ≤ b₃,

λi(JΨ4(Y )) ≥ b4, Ψ5(0) = 0, Ψ5(X) 6= 0 when X 6= 0, λi(Ψ5(Y )) ≥ b5. Then, Eq.(4) has no non-trivial periodic solution of whatever period.

Proof. To prove Theorem 2, we define a Lyapunov function V¹= V1(X, Y, Z, W, U ):

V1 = − Z 1

0

hΨ2(σZ)Z, Y i dσ − hU, Y i − Z 1

0

hσΨ3(σY )Y, Y i dσ

− Z 1

0

hΨ5(σX), Xi dσ − hAY, W i +1

2hAZ, Zi + hZ, W i . (10) It is easy to see from (10) that

V (0, 0, 0, 0, 0) = 0.

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and

V (0, 0, ε, ε, 0) ≥ 1

2hAε, εi + hε, εi

≥ 1

2(b1+ 1) kεk²> 0, for all arbitrary ε 6= 0, ε ∈ <ⁿ by the assumption λi(A) ≥ b1> 0.

Finally, let (X, Y, Z, W, U ) = (X(t), Y (t), Z(t), W (t), U (t)) be an arbitrary solution of system (6). Differentiating the Lyapunov function V1 with respect to the time t along this solution, we obtain

V˙1 = −d dt

Z 1 0

hΨ2(σZ)Z, Y i dσ − d dt

Z 1 0

hσΨ3(σY )Y, Y i dσ

−d dt

Z 1 0

hΨ5(σX), Xi dσ + hΨ2(Z)W, Y i

(11) + hΨ3(Y )Z, Y i + hΨ4(Y ), Y i + hΨ5(X), Y i + hW, W i .

It can be checked that d

dt Z 1

0

hΨ₂(σZ)Z, Y i dσ = Z 1

0

hΨ₂(σZ)Z, Zi dσ + Z 1

0

σ ∂

∂σhΨ₂(σZ)W, Y i dσ +

Z 1 0

hΨ₂(σZ)W, Y i dσ

= σ hΨ₂(σZ)W, Y i ¹₀ +

Z 1 0

hΨ₂(σZ)Z, Zi dσ

= hΨ2(Z)W, Y i + Z 1

0

hΨ2(σZ)Z, Zi dσ,

d dt

Z 1 0

hΨ3(σY )Y, Y i dσ = Z 1

0

σ ∂

∂σ hσΨ3(σY )Z, Y i dσ + Z 1

0

hσΨ2(σY )Z, Y i dσ

= σ²hΨ3(σY )Z, Y i

1

0 = hΨ3(Y )Z, Y i , d

dt Z 1

0

hΨ5(σX), Xi dσ = Z 1

0

σ hJΨ₅(σX)Y, Xi dσ + Z 1

0

hΨ5(σX), Y i dσ

= Z 1

0

σ ∂

∂σhΨ5(σX), Y i dσ + Z 1

0

hΨ5(σX), Y i dσ

= σ hΨ5(σX), Y i

¹₀ = hΨ5(X), Y i . Substituting the last three estimates into (11), we have

V˙1= hΨ4(Y ), Y i − Z 1

0

hΨ2(σZ)Z, Zi dσ + hW, W i .

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On the other hand, it is clear that Ψ₄(Y ) =

Z 1 0

J_Ψ₄(σ₁Y )Y dσ₁ so that

hΨ4(Y ), Y i =

Z 1 0

JΨ₄(σ1Y )Y dσ1, Y

≥ 1

2b4hY, Y i by λi(JΦ₄(Y )) ≥ b4> 0.

Then,

V˙1≥1

2b4hY, Y i − Z 1

0

hΨ2(σZ)Z, Zi dσ + hW, W i ≥ 0

by the assumptions of Theorem 2. The rest of the proof is similar to the proof of Theorem 1. Therefore, we omit the details of the poof.

References

[1] Ezeilo, J.O.C., ”Instability theorems for certain fifth-order differential equations”, Math.

Proc. Cambridge Philos. Soc., 84:2 (1978), 343-350.

[2] Ezeilo, J.O.C., ”Extension of certain instability theorems for some fourth and fifth order differential equations”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 66:8 (1979), no. 4, 239-242.

[3] Ezeilo, J.O.C., ”A further instability theorem for a certain fifth-order differential equation”, Math. Proc. Cambridge Philos. Soc., 86:3 (1979), 491-493.

[4] Horn, R.A., and Johnson, C.R., Topics in matrix analysis, Cambridge University Press, Cambridge, 1994.

[5] Krasovskii, N., ”On conditions of inversion of A. M. Lyapunov’s theorems on instability for stationary systems of differential equations (Russian)”, Dokl. Akad. Nauk. SSSR (N.S.), 101, (1955), 17-20.

[6] Li, W.J., and Duan, K.C., ”Instability theorems for some nonlinear differential systems of fifth order”, J. Xinjiang Univ. Natur. Sci. 17:3 (2000), 1-5, 18.

[7] Li, W.J., and Yu, Y.H., ”Instability theorems for some fourth-order and fifth-order differential equations”, (Chinese) J. Xinjiang Univ. Natur. Sci., 7:2 (1990), 7-10.

[8] Sadek, A.I., ”Instability results for certain systems of fourth and fifth order differential equations”, Appl. Math. Comput. 145:2-3 (2003), 541-549.

[9] Sun, W.J,, and Hou, X., ”New results about instability of some fourth and fifth order nonlinear systems”, (Chinese)J. Xinjiang Univ. Natur. Sci. 16:4 (1999), 14-17.

[10] Tejumola, H.O., ”Integral conditions of existence and non-existence of periodic solutions of some sixth and fifth order ordinary differential equations”, J. Nigerian Math. Soc., 31 (2012), 23-33.

[11] Tunc, C., ”On the instability of solutions of certain nonlinear vector differential equations of fifth order”, Panamer. Math. J., 14:4 (2004), 25-30.

[12] Tunc, C., ”An instability result for a certain non-autonomous vector differential equation of fifth order. Panamer”, Math. J., 15:3 (2005), 51-58.

[13] Tunc, C., ”Further results on the instability of solutions of certain nonlinear vector differential equations of fifth order”,Appl. Math. Inf. Sci. 2:1 (2008), 51-60.

[14] Tunc, C., ”Instability of solutions for certain nonlinear vector differential equations of fourth order”, Nelnn Koliv. 12 (2009), no. 1, 120-129; translation in Nonlinear Oscil. (N. Y.) 12:1 (2009), 123-132.

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[15] Tunc, C., ”Instability in multi delay functional differential equations of fourth order”, Fasc.

Math. 55:1 (2015), 189-198.

[16] Tunc, C., and Erdogan, F., ”On the instability of solutions of certain non-autonomous vector differential equations of fifth order”, SUT J. Math.43:1 (2007), 35-48.

[17] Tunc, C., and Karta, M., ”A new instability result to nonlinear vector differential equations of fifth order”,Discrete Dyn. Nat. Soc. 2008, Art. ID 971534, 6 pp.

[18] Tunc, C., and evli, H., ”On the instability of solutions of certain fifth order nonlinear differential equations”, Mem. Differential Equations Math. Phys. 35 (2005), 147-156.